Question

Using the binomial theorem , obtain the expansion of :


`(1+3x)^6+(1-3x)^6`

2. Using the binomial theorem , expand

`(1+2x)^5` , simplifying all the terms.

Answer 1

see explanation

Explanation:

Expand both factors and collect like term

Using Pascal' triangle with n = 6 to obtain the coefficients

1 6 15 20 15 6 1

Decreasing powers of 1 from
`1^(6)` to
`1^(0)`

Increasing powers of 3x from
`(3x)^(0)` to
`(3x)^(6)`

`1+3x)^(6)`

= 1.
`1^(6)`
`(3x)^(0)` + 6.
`1^(5)`
`(3x)^(1)` + 15.
`1^(4)`
`(3x)^(2)` + 20.
`1^(3)`
`(3x)^(3)` + 15.1²
`(3x)^(4)` + 6.
`1^(1)`
`(3x)^(5)` + 1.
`1^(0)`
`(3x)^(6)`

= 1 + 18x + 135x² + 540x³ + 1215
`x^(4)` + 1458
`x^(5)` + 729
`x^(6)`

--------------------------------------------------------------------------------------

`(1-3x)^(6)`

= 1.
`1^(6)`
`(-3x)^(0)` + 6.
`1^(5)`
`(-3x)^(1)` + 15.
`1^(4)`
`(-3x)^(2)` + 20.
`1^(3)`
`(-3x)^(3)` + 15.1²
`(-3x)^(4)` + 6.
`1^(1)`
`(-3x)^(5)` + 1.
`1^(0)`
`(-3x)^(6)`

= 1 - 18x + 135x² - 540x³ + 1215
`x^(4)` - 1458
`x^(5)` + 729
`x^(6)`

----------------------------------------------------------------------------------

Collecting like terms from both expressions

`(1+3x)^(6)` +
`(1-3x)^(6)`

= 2 + 270x² + 2430
`x^(4)` + 1458
`x^(6)`

----------------------------------------------------

(2)

Using Pascal's triangle with n = 5

1 5 10 10 5 1

Decreasing powers of 1 from
`1^(5)` to
`1^(0)`

Increasing powers of 2x from
`(2x)^(0)` to
`(2x)^(5)`

`(1+2x)^(5)`

= 1.
`1^(5)`
`(2x)^(0)` + 5.
`1^(4)`
`(2x)^(1)` + 10.
`1^(3)`
`(2x)^(2)` + 10.
`1^(2)`
`(2x)^(3)` + 5.
`1^(1)`
`(2x)^(4)`+ 1.
`1^(0)`
`(2x)^(5)`

= 1 + 10x + 40x² + 80x³ + 80
`x^(4)` + 32
`x^(5)`